FINE BEHAVIOR OF SYMBOLIC POWERS OF IDEALS
CRAIG HUNEKE AND MELVIN HOCHSTER
Abstract. A fundamental property connecting the symbolic powers and the usual powers
of ideals in regular rings was discovered by Ein, Lazarsfeld, and Smith in 2001, and later
extended by Hochster and Huneke in 2002. In this paper we give further generalizations
which give better results in case the quotient of the regular ring by the ideal is F-pure
or F-pure type. Our methods also give insight into a conjecture of Eisenbud and Mazur
concerning the existence of evolutions. The methods used come from tight closure and
reduction to positive characteristic.
This paper is written in appreciation of the many contributions of
Phil Griffith to commutative algebra.
1. Introduction
All given rings in this paper are commutative, associative with identity, and Noetherian.
In [3], L. Ein, R. Lazarsfeld, and K. Smith discovered the following remarkable fact about
the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0: if
c is the largest height1 of an associated prime of I, then I (cn) ⊆ I n for all n ≥ 0. Here, if W
is the complement of the union of the associated primes of I, I (t) denotes the contraction of
I t RW to R, where RW is the localization of R at the multiplicative system W . Their proof
depended on the theory of multiplier ideals (see [1], [17], [16], [23], and [24] for background),
including an asymptotic version, and, in particular, needed resolution of singularities as well
as vanishing theorems.
2000 Mathematics Subject Classification. Primary 13A35, 13B40, 13H05.
Key words and phrases. symbolic powers, regular ring, tight closure, F-pure.
Both authors were supported in part by grants from the National Science Foundation. The first author
was supported by DMS-0400633 and the second author was supported by grant DMS-0244405.
1See Discussion 1.1 and Theorem 2.1
1
2
CRAIG HUNEKE AND MELVIN HOCHSTER
Stronger results were obtained in [11, Theorem 1.1] with proofs by methods that were,
in some ways, more elementary. The results of [11] are valid in both equal characteristic
0 and in positive prime characteristic p, but depend on reduction to characteristic p. The
paper [11] used tight closure methods and, in consequence, needed neither resolution of
singularities nor vanishing theorems that may fail in positive characteristic.
In this paper we use ideas related to those of [11] to prove further results in this direction
that are more subtle.
E.g., in the local case the conclusion may be that a certain symbolic power of I is
contained in the product of the maximal ideal with a certain other symbolic power of I.
It appears to be very difficult to establish these theorems without tight closure theory or
some correspondingly intricate method: we do not know whether the techniques of [3] can
be used to obtain the main theorems here even in equal characteristic 0.
2
The papers [2], [12], and [13] study the existence of evolutions, which is equivalent to the
question of whether, for a prime P in a regular local ring (R, m), P (2) ⊆ mP . Eisenbud and
Mazur [2] ask whether this is always true in a regular local ring of equal characteristic 0.
Kunz gave a counterexample in characteristic 2 which is extended to all positive characteristics in [2]: these counterexamples are codimension three primes in regular local rings of
dimension four.3 Our Theorems 3.5 and 4.2 prove that in a regular local ring containing a
field, if P is a prime of codimension three, then P (4) ⊆ mP . In dimension 3 regular local
rings, codimension 2 primes are in the linkage class of a complete intersection (this is true
in regular local rings of any dimension whenever the quotient by the codimension two ideal
is Cohen-Macaulay), and the fact that P (2) ⊆ mP is established for primes in the linkage
class of a complete intersection in [2]. Our Theorems 3.5 and 4.2 prove that in a regular
local ring of arbitrary dimension containing a field, if P is a prime of codimension two, then
P (3) ⊆ mP . In general we are able to prove that if P is a prime of codimension c in a
regular local ring containing a field, then P (c+1) ⊆ mP (see Theorem 3.5).
2Since we finished this research, Shunsuke Takagi [26] has announced that he can obtain similar results,
and in some cases stronger results, using an interpretation of multiplier ideals via tight closure.
3Specifically, map K[[x , x , x , x ]] K[[t]] by sending x , x , x , x to ta , tb , tc , td resp. Let P be
1
2
3
4
1
2
3
4
the kernel. If K has characteristic p > 0 and the values of a, b, c, d are p2 , p(p + 1), p(p + 1) + 1, and
(p + 1)2 , resp., then f = xp+1
x2 − xp+1
− x1 xp3 + xp4 ∈ P (2) \ mP .
1
2
FINE BEHAVIOR OF SYMBOLIC POWERS OF IDEALS
3
We note that to prove the most basic form of our results, all that we need to know about
tight closure is the definition and the fact that, in a regular ring, every ideal is tightly closed.
The results of Section 4 require more of the theory, and we refer the reader to [14], [7]-[9],
[10], and [22] for further background on tight closure theory.
Although all of our proofs initially take place in positive characteristic, relatively standard
methods show that whenever, roughly speaking, the statements “make sense,” corresponding results hold for rings containing a field of characteristic 0. We need a definition before
stating our results.
Discussion 1.1. Our results are typically expressed in terms of a number associated with
I which we call the key number of I. We shall give here several possible definitions of this
term: the reader may choose any one of them. Our reason for proceeding in this way is that
the results are sharpest for a rather technical version of “key number” based on analytic
spread, but are correct and a lot less technical if one simply uses instead the largest number
of generators after localizing at an associated prime of the ideal (or the largest height of
an associated prime of the ideal). We note that all of the definitions that we give agree for
radical ideals in regular rings.
Thus, in the sequel, the reader may use any of the following as the definition of the key
number of I: the number obtained will, in general, depend on which definition is used, but
the theorems will be correct for any of these numbers (or any larger number).
(1) The key number of the ideal I is the largest number of generators of IRP for any
associated prime P of I.
(2) The key number of the ideal I is the largest height (or codimension) of any associated
prime P of I, where the height of P is the Krull dimension of RP .
4
CRAIG HUNEKE AND MELVIN HOCHSTER
(3) The key number of the ideal I is the largest analytic spread4 of IRP for any associated
prime P of I.
Because the analytic spread of I in a local ring R is bounded both by the Krull dimension
of R and by the least number of generators of I, the notion of key number given in (3) is
never larger than either of the other two. As already mentioned, the three notions coincide
in the case of a radical ideal of a regular ring.
We note that by the main results of [11], if c is the key number for I, then I (cn) ⊆ I n for
all nonnegative integers n: see Theorem 2.1 of Section 2. On first perusal of this paper the
reader may well want to focus on the case where I is radical or even prime.
It is a natural question to ask whether the result that I (cn) ⊆ I n for all n ≥ 1 can be
improved, perhaps by assuming more about the singularities of R/I. The following example,
which we learned from L. Ein, shows that one will not be able to improve this result too
much.
Example 1.2. In K[x1 , . . . , xn ], consider the primes (xi , xj ) for j 6= i. Let I denote their
intersection, which is generated by all monomials consisting of products of n − 1 of the
variables. This is a radical ideal of pure height 2 with key number c = 2.
For every integer k, xk1 · · · xkn ∈ I (2k) , but if k < n − 1, it is not in I k+1 : since each
generator omits one variable, for a product of k + 1 < n generators there must be some
variable that occurs in all k + 1 factors, and that variable will have an exponent of at least
k + 1 in the product. Note that the main result of [11] implies that I (2k) ⊆ I k .
In giving proofs in Section 3 we first give the argument in the case where the key number
c is defined as in (1). We then later explain the modifications in the arguments needed for
case (3). The point in case (3) is that by the introduction of an additional fixed multiplier,
we can, in essence, replace I, in certain localizations, by an ideal with c generators on which
4For a discussion of analytic spread and related ideas we refer the reader to [25], and [19]. A summary of
what is needed here is given in [11], §2.3. We note that when I ⊆ (R, m) with (R, m) local, the analytic spread
of I is the same as the Krull dimension of the ring (R/m) ⊗R grI R, where grI R = R/I ⊕ I/I 2 ⊕ I 2 /I 3 ⊕ · · ·
is the associated graded ring of R with respect to the I-adic filtration. When R/m is infinite, this is the
same as the smallest number of generators of an ideal J ⊆ I such that I is contained in the integral closure
of the ideal J.
FINE BEHAVIOR OF SYMBOLIC POWERS OF IDEALS
5
it is integrally dependent. We do not need to give an argument for the case where the key
number is defined as in (2), since the number defined in (3) is never larger.
The result that follows, Theorem 1.3, is a composite of Theorems 3.5, 3.6, and 4.2 containing our main results.
Theorem 1.3. Let (R, m) be a regular local ring containing a field and let I be a proper
ideal of R with key number c.
(1) For every s ≥ 1, I (cs+1) ⊆ mI (s) .
(2) If R is of equal characteristic p and R/I is F -pure, then I (rc−1) ⊆ II (r−1) .
We shall also discuss a version of part (2) for local rings of finitely generated algebras
over a field of characteristic 0 in Section 3: one needs to replace the notion of “F -pure” by
a suitable characteristic 0 notion (“F -pure type”).
We do not know how to prove (1) or (2) by elementary means, even when (R, m) is local
and has dimension 3, s = 1, and I = P is a prime of codimension 2: in that case (1) gives
that P (3) ⊆ mP .
Our results take a simpler form when s = 1 (in (1)) or r = 2 (in (2)) because I (1) = I,
and we make this explicit:
Theorem 1.4. Let (R, m) be a regular local ring containing a field and let I be a proper
ideal of R with key number c.
(1) I (c+1) ⊆ mI.
(2) If (R, m) is local of equal characteristic p and R/I is F -pure, then I (2c−1) ⊆ I 2 .
In the next section we recall the main results of [11]. Section 3 contains the proofs of our
results for regular rings in positive prime characteristic, and Section 4 contains the equal
characteristic 0 versions of our results.
2. Prior comparison results
The main results of [11] in all characteristics are summarized in the following theorem.
Note that I ∗ denotes the tight closure of the ideal I. The characteristic zero notion of tight
6
CRAIG HUNEKE AND MELVIN HOCHSTER
closure used in this paper is the equational tight closure of [10] (see, in particular Definition
(3.4.3) and the remarks in (3.4.4) of [10]). This is the smallest of the characteristic zero
notions of tight closure, and therefore gives the strongest result.
Theorem 2.1. Let R be a Noetherian ring containing a field. Let I be any ideal of R, and
let c be the key number5 of I.
(1) If R is regular, I (cn+kn) ⊆ (I (k+1) )n for all positive n and nonnegative k. In particular, I (cn) ⊆ I n for all positive integers n.
(2) If I has finite projective dimension then I (cn) ⊆ (I n )∗ for all positive integers n.
(3) If R is finitely generated, geometrically reduced (in characteristic 0, this simply
means that R is reduced) and equidimensional over a field K, and locally I is either
0 or contains a nonzerodivisor (this is automatic if R is a domain), then, with J
equal to the Jacobian ideal of R over K, for every nonnegative integer k and positive
integer n, we have that J n I (cn+kn) ⊆ ((I (k+1) )n )∗ and J n+1 I (cn+kn) ⊆ (I (k+1) )n . In
particular, we have that J n I (cn) ⊆ (I n )∗ and J n+1 I (cn) ⊆ I n for all positive integers
n.
The following result is implicit but not explicit in [11].
Theorem 2.2. Let R be a regular ring of positive prime characteristic p. Let I be an ideal
of R.
(1) Let c be the largest number of generators of I after localizing at an associated prime
of I. Then I (cnq) ⊆ (I (n) )[q] for every n ∈ N and q = pe .
(2) Let c be the the key number of I in any of the senses of Discussion 1.1. Then there
is a positive integer s such that I s I (cnq) ⊆ (I (n) )[q] for every n ∈ N and q = pe .
Proof. Let W be the multiplicative system that is the complement of the union of the
associated primes of I. The elements of W are not zerodivisors on any symbolic power of
of I, by construction of the symbolic powers, nor on any bracket power of a symbolic power
of I, since the Frobenius endomorphism is flat: see, for example, Lemma 2.2(d) of [11] (and
[20], [15], [5] for several related results).
5See Discussion 1.1
FINE BEHAVIOR OF SYMBOLIC POWERS OF IDEALS
7
(1) Thus, it will suffice to show that I (cnq) RW ⊆ (I (n) )[q] RW , i.e., that I cnq RW ⊆
(I n )[q] RW . Since RW is semilocal, it suffices to show this after localizing at a maximal
ideal, and this will be a (maximal) associated prime P of I. Thus, we need only show that
[q]
(IP )cnq ⊆ (IP )n . Since IP has at most c generators, say f1 , . . . , fc (we may take some
of these to be 0), by definition of c, any monomial of degree cnq in these must be such that
[q]
at least one of the fj has exponent ≥ nq, which means that it is in (IP )n , and the result
follows.
(2) We replace R by R[t] where t is an indeterminate, and I by its expansion to this ring.
The new associated primes of I are the expansions of the original associated primes. The
crucial effect of this trick is that now the localization of the ring at any associated prime of
the ideal has infinite residue field. We go back to our original notation and call the ring R.
Then for each associated prime P of I we may choose an ideal J of RP with c generators
such that IRP is integral over J. Then there is an integer sP such that I sP I N RP ⊆ J N
for every N ∈ N , Choose s to be at least the maximum of these finitely many sP . We
shall show that I s I (cnq) ⊆ (I (n) )[q] . Since no element of W is a zerodivisor on on (I (n) )[q] , it
suffices to show this after localizing at W and hence after localizing at each of the associated
primes P of I. We replace R by RP and I by IRP . But then all we need to show is that
I s I cnq ⊆ (I n )[q] , and we have that I s I cnq ⊆ J cnq ⊆ (J n )[q] (exactly as in part (1), since J
has c generators), and this is contained in (I n )[q] .
3. The main results for characteristic p regular rings
Let R be a regular domain of positive characteristic p > 0, let I be an ideal of R, let c be
the key number6 for I, and let r be an integer with r ≥ 2. We define a sequence of ideals
as follows:
Definition 3.1. Fix an integer k, 0 ≤ k ≤ r(c − 1). Set I0,k = I, and inductively set
In,k = In−1 I (r−1) : I (rc−k) . Set Jk = ∪j Ij,k . We will usually simplify notation when k is
fixed and write In = In,k .
6See Discussion 1.1
8
CRAIG HUNEKE AND MELVIN HOCHSTER
Observe that Ij form an increasing sequence of ideals (depending on k). Since k ≤ r(c−1)
it follows that rc − k ≥ rc − r(c − 1) = r and thus In−1 I (rc−k) ⊆ In−1 I (r−1) , and In−1 ⊆
In−1 I (rc−k) : I (r−1) = In . Hence Jk = IN for all large N .
Theorem 3.2. Let R be a regular domain of positive characteristic p > 0, let I be an ideal
of R, let c be the key number7 for I, and let r be an integer with r ≥ 2. For all q = pe , and
any integer k, 0 ≤ k ≤ r(c − 1),
[q]
I (q((n+1)k−nc)) ⊆ In+1 .
Proof. We first assume that key number is defined as in Discussion 1.1, (1) and prove the
case n = 0. We need to prove that
I (kq) ⊆ II (r−1) : I (rc−k)
Since we are in a regular ring, II (r−1) : I (rc−k)
[q]
[q]
.
= (II (r−1) )[q] : (I (rc−k) )[q] . Suppose
that u ∈ I (qk) and v ∈ I (rc−k) . It will suffice to show that uv q ∈ (II (r−1) )[q] . Evidently, we
may assume that v 6= 0.
[Q]
Say b ≥ rc − k. Fix any power Q of p. We shall show that v b (uv q )Q ∈ (II (r−1) )[q]
for
∗
all Q. Since b is independent of Q, this shows that uv q ∈ (II (r−1) )[q] (the tight closure),
and since R is regular, this will establish that uv q ∈ (II (r−1) )[q] .
Fix Q. By the division algorithm, qQ = (a − 1)(rc − k) + ρ, where where a − 1 ∈ N and
0 ≤ ρ < rc − k. Then ρ + b ≥ rc − k Clearly
(∗)
a(rc − k) ≥ qQ.
Then
v b (uv q )Q = uQ v b+qQ ∈ (uQ v ac v a
(r−1)c−k
) = (v ac )(uQ v a((r−1)c−k) ).
Now, since v ∈ I (rc−k) , we have that
v ac ∈ I ((rc−k)ac) ⊆ I (qQc) ⊆ I [qQ]
by Theorem 2.2, and since u ∈ I (qk) we find that
uQ v a((r−1)c−k) ⊆ I (qQk+(rk−c)a((r−1)c−k)) ,
7See Discussion 1.1
FINE BEHAVIOR OF SYMBOLIC POWERS OF IDEALS
9
and, using (∗), the exponent is ≥ qQk + qQ (r − 1)c − k = qQ(r − 1)c. Thus,
uQ v a((r−1)c−k) ⊆ I (qQ(r−1)c) ⊆ (I (r−1) )[qQ]
by Theorem 2.2. Multiplying, we have that
v b (uv q )Q ∈ I [qQ] (I (r−1) )[qQ] = (II (r−1) )[qQ] = (II (r−1) )[q]
[Q]
,
as required.
We next describe the changes needed in the argument in the case where the key number
is defined as in Discussion 1.1. If I = (0) there is nothing to prove. If not, choose s as
in Theorem 2.2(b) and let y be any nonzero element of I. The argument is almost the
[Q]
same, but we show instead that y 2s v b (uv q )Q ∈ (II (r−1) )[q]
for all Q, which again allows
us to conclude that uv q ∈ (II (r−1) )[q] )∗ . In the final paragraph of the argument we get
that y s v ac ⊆ y s I (qQc) ⊆ I [qQ] by Theorem 2.2(b), and, similarly, that y s uQ v a((r−1)c−k) ⊆
I (qQ(r−1)c) ⊆ (I (r−1) )[qQ] by Theorem 2.2(b). We can multiply: the argument is otherwise
unchanged. This completes the case n = 0.
We now assume the result for n − 1 and prove it for n. This induction works for every
choice of key number as defined in Discussion 1.1.
We need to prove that
[q]
I (q((n+1)k−nc)) ⊆ In+1 = (In I (r−1) : I (rc−k) )[q] .
Suppose that u ∈ I (q((n+1)k−nc) and v ∈ I (rc−k) . It will suffice to show that uv q ∈
(In I (r−1) )[q] . Evidently, we may assume that v 6= 0.
First suppose that c ≤ k. Then (n + 1)k − nc = k + n(k − c) ≥ k and the result follows
from the case n = 0, since by that case
I (q((n+1)k−nc)) ⊆ I (qk) ⊆ (II (r−1) : I (rc−k) )[q] ⊆ (In I (r−1) : I (rc−k) )[q] .
We can therefore assume that c > k. Fix any power Q of p. By the division algorithm,
(c − k)qQ = (a − 1)(rc − k) + ρ,
[Q]
where where a − 1 ∈ N and 0 ≤ ρ < rc − k. We shall show that v ρ (uv q )Q ∈ (In I (r−1) )[q]
∗
for all Q. Since ρ is independent of Q, this shows that uv q ∈ (In I (r−1) )[q] (the tight
10
CRAIG HUNEKE AND MELVIN HOCHSTER
closure), and since R is regular, this will establish that uv q ∈ (In I (r−1) )[q] . Note that
a(rc − k) ≥ (c − k)qQ. We break the product into two terms:
v ρ (uv q )Q = (uQ v a ) · v ρ+qQ−a .
Now, since v ∈ I (rc−k) , we have that
v a ∈ I ((rc−k)a) ⊆ I ((c−k)qQ)
and hence
uQ v a ∈ I ((c−k)qQ) I (qQ((n+1)k−nc)) ⊆ I (qQ(nk−(n−1)c) ⊆ In[q] .
The last step follows from our induction.
Observe that
(ρ+qQ−a)(rc−k) ≥ qQ(rc−k)−((a−1)(rc−k)+ρ)) = qQ(rc−k)−(c−k)qQ = qQ(r−1)c
so that
v ρ+qQ−a ∈ I ((r−1)cqQ) ⊆ (I (r−1) )[qQ]
by Theorem 2.2.
Hence
v ρ (uv q )Q = (uQ v a ) · v ρ+qQ−a ∈ (In I (r−1) )[qQ]
which proves that uv q ∈ (In I (r−1) )∗ and finishes the proof of the theorem.
Corollary 3.3. Let (R, m) be a regular local ring of positive characteristic p > 0, let I be
an ideal of R, and let c be the key number8 Then either I (rc−1) ⊆ II (r−1) for every r ≥ 2
or I (q) ⊆ m[q] for every q = pe .
Proof. take k = 1 and apply Theorem 3.2 to the ideal I1 . Either I1 = R, in which case
[q]
I (rc−1) ⊆ II (r−1) , or else it is a proper ideal, in which case I (q) ⊆ I1 ⊆ m[q] .
Remark 3.4. The second of the alternative conclusions is a very strong form of the ZariskiNagata theorem9 which asserts that P (N ) ⊆ mn for every prime of the regular local ring
8See Discussion 1.1
9The theorem is equivalent to Theorem 1 of [6], where Zariski’s proof is given; for Nagata’s proof see [18,
p. 143].
FINE BEHAVIOR OF SYMBOLIC POWERS OF IDEALS
11
(R, m). For q = pe , m[q] tends to be quite a bit smaller than mq . Note that the second of
the alternate conclusions does imply10 that the Zariski-Nagata theorem holds even when N
is not a power of p.
Theorem 3.5. Let (R, m) be a regular local ring of positive characteristic p, let I be an
ideal of R, and let c be the key number for I. For all s ≥ 1,
I (cs+1) ⊆ mI (s) .
Proof. We claim that Jc−1 = R. By Theorem 3.2,
[q]
I (q((n+1)(c−1)−nc)) ⊆ In+1
for all n. But for all large n, (n + 1)(c − 1) − nc = c − n + 1 < 0, and hence In+1 = R. Since
In+1 ⊆ Jc−1 we obtain Jc−1 = R.
Choose n least with the property that In = R (for k = c − 1). Note that I0 = I 6= R, so
that In−1 6= R. But then R = In = In−1 I (r−1) : I (rc−c+1) implies that
I (rc−c+1) ⊆ In−1 I (r−1) ⊆ mI (r−1) .
Setting s = r − 1 gives the conclusion of the theorem.
Theorem 3.6. Let R be a regular ring of positive characteristic p > 0, let I be a proper
ideal of R, let r ≥ 2 be an integer, and let c be the key number for I. If R/I is F -pure, then
I (rc−1) ⊆ II (r−1) .
Proof. The proof reduces at once to the local case, so we may assume that (R, m) is a regular
local ring. If I (c−1) ⊆ II (r−1) we are done, since rc − 1 ≥ c − 1. Thus, we may assume that
I (c−1) * II (r−1) .
By Fedder’s criterion for F -purity (cf. [4]), I [q] : I * m[q] . If I (rc−1) * II (r−1) , then
we may set k = 1 in Theorem 3.2. But then I (kq) = I (q) ⊆ (II (r−1) : I (rc−1) )[q] ⊆ m[q] , a
contradiction, for then I (q) ⊆ m[q] , while I [q] : I * m[q] .
10Suppose u ∈ P (N ) . For any q ≥ N we have q = aN + b, where a = b q c and 0 ≤ b < N . Then
N
ub ua ∈ P (q) ⊆ mq , which shows that (b+a) ordm u ≥ N a+b for all q ≥ N , and so ordm u ≥ (N a+b)/(b+a) =
(N + ab )/(1 + ab ). Since b < N while a−→∞ as q−→∞, this is > N − 1 for q 0. Thus, ordm u ≥ N , i.e.,
u ∈ mN .
12
CRAIG HUNEKE AND MELVIN HOCHSTER
4. The main results in characteristic 0
In this section we give the extensions of the various positive characteristic results to the
equal characteristic case. As mentioned in Section 2, the notion of tight closure that we use
here is that of equational tight closure from [10, Sections 3.4.3–4]. The main results of this
section are contained in Theorem 4.2 below. We need to do some groundwork before we can
prove that theorem, however. The proof of the main results depends on three steps: one is
to localize and complete, the second is to descend from the complete case to the affine case,
and the third is to use reduction to positive characteristic in the affine case. We follow the
same path as was done in detail in [11]. To state the main theorem we need the definition
of F -pure type.
Definition 4.1. Let R be a ring which is finitely generated over a field k of characteristic
0. The ring R is said to be of F -pure type if there exists a finitely generated Z-algebra
A ⊆ k and a finitely generated A-algebra RA which is free over A such that R ∼
= RA ⊗A K
and such that for all maximal ideals µ in a Zariski dense subset of Spec(A), with κ = A/µ,
the fiber rings RA ⊗A κ are F-pure.
Theorem 4.2. Let R be a regular ring containing a field of characteristic 0, and let I be a
proper ideal of R with key number c.
(1) If R is local with maximal ideal m, then for all s ≥ 1,
I (cs+1) ⊆ mI (s) .
(2) If R is of finite type over a field of equal characteristic 0 and R/I is of F -pure type,
then I (rc−1) ⊆ II (r−1) .
Proof. We prove a stronger statement to prove (1) which does not require the ring to be
local. Recall the definition of Ij , which makes sense in all characteristics: Fix an integer k,
0 ≤ k ≤ r(c − 1). Set I0 = I, and inductively set In = In−1 I (r−1) : I (rc−k) . Set Jk = ∪j Ij .
Recall that Ij form an increasing sequence of ideals (depending on k), since In−1 I (rc−k) ⊆
In−1 I (r−1) as k ≤ (r − 1)c, so that rc − k ≥ rc − r(c − 1) = r. Hence Jk = IN for all large
N.
FINE BEHAVIOR OF SYMBOLIC POWERS OF IDEALS
13
We replace (1) by
(10 ): Jc−1 = R.
We first prove that (10 ) implies (1). Assume (10 ) and assume that R is local with maximal
ideal m. The proof is entirely similar to the proof of Theorem 3.5 in positive characteristic.
We let k = c − 1, and consider the ascending chain of ideal {Ij }. By (10 ), for some large n,
In = R. Choose n least with this property. Note that I0 = I 6= R, so that In−1 6= R and is
a proper ideal. But then R = In = In−1 I (r−1) : I (rc−c+1) implies that
I (rc−c+1) ⊆ In−1 I (r−1) ⊆ mI (r−1) .
Setting s = r − 1 gives (1).
We prove (10 ) for finitely generated algebras over a field K of characteristic 0, and at the
same time prove (2). Assume there is a counterexample in either of these cases. We use the
standard descent theory of Chapter 2 of [10] to replace the field K by a finitely generated
Z-subalgebra A, so that we have a counterexample in an affine algebra RA over A with
RA ⊆ R and R ∼
= K ⊗A RA . In particular, RA will be reduced. In doing so we descend I to
an ideal IA of RA as well as the ideals and their prime radicals in its primary decomposition.
We define the ideals IjA and JkA in a similar manner. In case (10 ), we have the stable ideal
JkA which is the union of the ideals IjA and we are assuming that 1 ∈
/ J(c−1)A . In case (3)
we have an element uA that fails to satisfy the containment we are trying to prove. Since R
is regular, we can localize at a nonzero element of A to make RA smooth over A. In either
case, we can localize at a nonzero element of A to make A smooth over Z.
For variable maximal ideals of A, we denote the residue field by κ. Notice that as we pass
to fibers κ−→Rκ = RA ⊗A κ we may assume that each minimal prime PA of IA becomes a
radical ideal whose minimal primes in Rκ are all of the same height as the original. Thus,
in the fiber, the primary decomposition of Iκ may have more components, but each of these
will be obtained from the image of one of the original components by localization. The
biggest analytic spread after localizing at an associated prime will not change. The ring Rκ
will be regular, and in part (2) we know that for a dense set of maximal ideals of A, the
fiber Rκ /Iκ is F-pure. Both (2) and (10 ) now follow since both results hold in characteristic
p for for a dense set of closed fibers.
14
CRAIG HUNEKE AND MELVIN HOCHSTER
We now consider the general case for (1). The problem reduces to the local case, and then
b may have more
it suffices to prove the required containment after completion. Although I R
associated primes, Discussion 1.1 shows that the biggest analytic spread as one localizes at
these cannot increase.
b so that (Ib(j) )n = (I (j) )n R.
b
Since R is regular note that for all integers j, Ib(j) = I (j) R,
b (j) R
b are among those associated to R/P
b R
b for some associated
(The associated primes of R/I
prime P of I, by Proposition 15 in IV B.4. of [21], since any associated prime of I (j) must
be an associated prime of I, and by another application of Proposition 15 in IV B.4. of [21]
b Thus,
these in turn are associated primes of I R.)
b ∩ R = mI (s) .
b Ib(s) ∩ R = mI (s) R
I (cs+1) ⊆ Ib(cs+1) ⊆ m
b over R.
as required, by the faithful flatness of R
We may now use Theorem 4.3 of [11] to descend to a suitable affine algebra over a
coefficient field for the complete local ring, and the results follow from what we have already
proved in the affine case. This reduction is entirely similar to the reduction done in [11],
and we refer the reader to that paper for full details.
Remark 4.3. We still do not know the best possible conclusion, even in codimension two.
For example, in K[x, y, z], for homogeneous primes P1 , . . . , Ps of height two, we do not
know whether or not (P1 ∩ · · · ∩ Ps )(3) ⊆ (P1 ∩ · · · ∩ Ps )2 . All evidence suggests this is
always true, but we have not been able to extend our methods to decide this question.
It is also possible that for regular local rings (R, m) in all characteristics, I (cs) ⊆ mI,
where c is the key number for I. If so, this would prove the Eisenbud-Mazur conjecture
(even in positive characteristic) for prime ideals of codimension two. It would also show
that the counterexamples in positive characteristic are sharp.
References
1. L. Ein and R. Lazarsfeld, A geometric effective Nullstellensatz, Invent. Math., 137 (1999), 427–448.
FINE BEHAVIOR OF SYMBOLIC POWERS OF IDEALS
15
2. D. Eisenbud and B. Mazur, Evolutions, symbolic squares, and Fitting ideals, J. Reine Angew. Math.
488 (1997), 189–201.
3. L. Ein, R. Lazarsfeld, and K. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent.
Math. 144 (2001), 241–252.
4. R. Fedder, F-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461–480.
5. J. Herzog, Ringe der Characteristik p und Frobenius-funktoren, Math Z. 140 (1974), 67–78.
6. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann.
of Math., 79 (1964), 205–326.
7. M. Hochster and C. Huneke Tight closure and strong F -regularity, Mémoires de la Société Mathématique
de France, 38 (1989), 119–133.
8. M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer.
Math. Soc., 3 (1990), 31–116.
9. M. Hochster and C. Huneke, F-regularity, test elements, and smooth base change, Trans. Amer. Math.
Soc., 346 (1994), 1–62.
10. M. Hochster and C. Huneke, Tight closure in equal characteristic zero, preprint, available at
http://www.math.lsa.umich.edu/ hochster/msr.html.
11. M. Hochster and C. Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math., 147
(2002), 349–369.
12. R. Hübl, Evolutions and valuations associated to an ideal, Jour. Reine Angew. Math., 517 (1999),
81–101.
13. R. Hübl and C. Huneke, Fiber cones and the integral closure of ideals, Collect. Math. 52 (2001), 85–100.
14. C. Huneke, Tight Closure and Its Applications, Proc. of the CBMS Conference held at Fargo, North
Dakota, 1995, C.B.M.S. Regional Conference Series, Amer. Math. Soc. Providence, R.I. 88 (1996).
15. E. Kunz, Characterizations of regular local rings of characteristic p, Amer. J. Math. 91 (1969), 772–784.
16. R. Lazarsfeld, Positivity in Algebraic Geometry, Springer Verlag, (2000).
17. J. Lipman, Adjoints of ideals in regular local rings, Math. Res. Lett. 1 (1994), 739–755.
18. M. Nagata, Local Rings, Interscience, New York (1962).
19. D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. of the Cambridge Phil. Soc. 50
(1954), 145–158.
20. C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Publ. Math. I.H.E.S. (Paris)
42 (1973), 323–395.
21. J.-P. Serre, Algèbre local · Multiplicité Lecture Notes in Mathematics 11, Springer-Verlag, Berlin, Heidelberg, New York (1965).
22. K. E. Smith, Tight closure of parameter ideals, Invent. Math. 115 (1994), 41–60.
16
CRAIG HUNEKE AND MELVIN HOCHSTER
23. K. E. Smith, The multiplier ideal is a universal test ideal, Communications in Algebra 28 (2000), 5915–
5929.
24. I. Swanson, Linear equivalence of topologies, Math. Z. 234 (2000), 755–775.
25. I. Swanson and C. Huneke, Integral Closure of Ideals, Rings and Modules, book, to appear, (2005).
26. S. Takagi, A subadditivity formula for multiplier ideals on singular varieties, Extended abstracts of the
workshop ”Kommutative Algebra” Oberwolfach Rep. 2 (2005), 1073–1126.
Department of Mathematics, University of Kansas, Lawrence, KS 66045
E-mail address: [email protected]
URL: http://www.math.ku.edu/~huneke
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
E-mail address: [email protected]
URL: http://www.math.lsa.umich.edu/~hochster